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Rebounding hydrogel sphere water entry

Published online by Cambridge University Press:  13 June 2025

Xiaoyan Ye Open the ORCID record for Xiaoyan Ye [Opens in a new window] ,
Kaizhao Qin ,
Rui Yang Open the ORCID record for Rui Yang [Opens in a new window]  and
Joshua A. Dijksman Open the ORCID record for Joshua A. Dijksman [Opens in a new window]
Xiaoyan Ye*
Affiliation:
Key Laboratory of Mechanics on Disaster and Environment in Western China attached to the Ministry of Education of China, and Department of Mechanics and Engineering Science, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
Kaizhao Qin
Affiliation:
Key Laboratory of Mechanics on Disaster and Environment in Western China attached to the Ministry of Education of China, and Department of Mechanics and Engineering Science, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, PR China
Rui Yang
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Joshua A. Dijksman
Affiliation:
Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
*
Corresponding author: Xiaoyan Ye, yexy@lzu.edu.cn
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Abstract

The interaction between elastic structures and fluid interfaces, known as ‘hydroelastic’ problems, presents unique challenges to classical frameworks established for rigid spheres and liquid droplets. In this work, we experimentally demonstrate an intriguing phenomenon where ultrasoft hydrogel spheres rebound from a water surface at high impact speeds, even when their density exceeds that of water. We further propose a theoretical force-balance model, incorporating energy redistribution and potential flow theory, to predict the critical impact speed for the transition from sinking to rebounding, as well as the temporal evolution of both spreading diameter and cavity expansion. Our findings extend the classical Weber- and Bond-number-dominated paradigms for rigid spheres and liquid droplets, demonstrating that hydrogel dynamics is controlled by a modified elastocapillary Mach number, with rebound achievable even for hydrophilic spheres. These findings improve the understanding of soft-impact hydrodynamics and offer design principles for applications in biomimetic robotics and energy-absorbing materials.

JFM classification

Aerodynamics: Flow-structure interactions Drops and Bubbles: Drops and Bubbles Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1013 , 25 June 2025 , A2
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Aristoff, J.M. & Bush, J.W. 2009 Water entry of small hydrophobic spheres. J. Fluid Mech. 619, 45–78.10.1017/S0022112008004382CrossRefGoogle Scholar
Aristoff, J.M., Truscott, T.T., Techet, A.H. & Bush, J.W. 2010 The water entry of decelerating spheres. Phys. Fluids 22 (3), 032102.10.1063/1.3309454CrossRefGoogle Scholar
Arora, S., Fromental, J.M., Mora, S., Phou, T., Ramos, L. & Ligoure, C. 2018 Impact of beads and drops on a repellent solid surface: a unified description. Phys. Rev. Lett. 120 (14), 148003.10.1103/PhysRevLett.120.148003CrossRefGoogle ScholarPubMed
Belden, J., Hurd, R.C., Jandron, M.A., Bower, A.F. & Truscott, T.T. 2016 Elastic spheres can walk on water. Nat. Commun. 7 (1), 10551.10.1038/ncomms10551CrossRefGoogle ScholarPubMed
Benusiglio, A., Quéré, D. & Clanet, C. 2014 Explosions at the water surface. J. Fluid Mech. 752, 123–139.10.1017/jfm.2014.255CrossRefGoogle Scholar
Bush, J.W. & Hu, D.L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38 (1), 339–369.10.1146/annurev.fluid.38.050304.092157CrossRefGoogle Scholar
Chantelot, P., Coux, M., Clanet, C. & Quéré, D. 2018 Drop trampoline. Europhys. Lett. 124 (2), 24003.10.1209/0295-5075/124/24003CrossRefGoogle Scholar
Cheng, N.-S. 2008 Formula for the viscosity of a glycerol- water mixture. Ind. Engng Chem. Res. 47 (9), 3285–3288.10.1021/ie071349zCrossRefGoogle Scholar
Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 1–19.10.1017/S0022112007007343CrossRefGoogle Scholar
Gilet, T. & Bush, J.W. 2012 Droplets bouncing on a wet, inclined surface. Phys. Fluids 24 (12), 122103.10.1063/1.4771605CrossRefGoogle Scholar
Glasheen, J. & McMahon, T. 1996 A hydrodynamic model of locomotion in the basilisk lizard. Nature 380 (6572), 340–342.10.1038/380340a0CrossRefGoogle Scholar
Halswell, P., Wilson, P., Taunton, D. & Austen, S. 2012 Hydroelastic inflatable boats: relevant literature and new design considerations. Intl J. Small Craft Technol. 154, B39.Google Scholar
Hsiao, M., Lichter, S. & Quintero, L.G. 1988 The critical weber number for vortex and jet formation for drops impinging on a liquid pool. Phys. Fluids 31 (12), 3560–3562.10.1063/1.866872CrossRefGoogle Scholar
Hurd, R.C., Belden, J., Jandron, M.A., Fanning, D.T., Bower, A.F. & Truscott, T.T. 2017 Water entry of deformable spheres. J. Fluid Mech. 824, 912–930.10.1017/jfm.2017.365CrossRefGoogle Scholar
Korobkin, A. & Pukhnachov, V. 1988 Initial stage of water impact. Annu. Rev. Fluid Mech. 20 (1), 159–185.10.1146/annurev.fl.20.010188.001111CrossRefGoogle Scholar
Kubota, Y. & Mochizuki, O. 2015 Splash formation due to a frog diving into water. World J. Mech. 5 (07), 129–137.10.4236/wjm.2015.57014CrossRefGoogle Scholar
Lee, D.-G. & Kim, H.-Y. 2008 Impact of a superhydrophobic sphere onto water. Langmuir 24 (1), 142–145.10.1021/la702437cCrossRefGoogle ScholarPubMed
Lhuissier, H., Sun, C., Prosperetti, A. & Lohse, D. 2013 Drop fragmentation at impact onto a bath of an immiscible liquid. Phys. Rev. Lett. 110 (26), 264503.10.1103/PhysRevLett.110.264503CrossRefGoogle ScholarPubMed
Lohse, D., Bergmann, R., Mikkelsen, R., Zeilstra, C., Van Der Meer, D., Versluis, M., Van Der Weele, K., van der Hoef, M. & Kuipers, H. 2004 Impact on soft sand: void collapse and jet formation. Phys. Rev. Lett. 93 (19), 198003.10.1103/PhysRevLett.93.198003CrossRefGoogle ScholarPubMed
MotMottt, R.L., Noor, F.M. & Aziz, A.A. 2006 Applied fluid mechanics.Google Scholar
Rabbi, R., Kiyama, A., Allen, J.S. & Truscott, T. 2022 Droplet lift-off from hydrophobic surfaces from impact with soft-hydrogel spheres. Commun. Phys. 5 (1), 331.10.1038/s42005-022-01114-8CrossRefGoogle Scholar
Richard, D. & Quéré, D. 2000 Bouncing water drops. Europhys. Lett. 50 (6), 769–775.10.1209/epl/i2000-00547-6CrossRefGoogle Scholar
Richardson, E.G. 1948 The impact of a solid on a liquid surface. Proc. Phys. Soc. 61 (4), 352–367.10.1088/0959-5309/61/4/308CrossRefGoogle Scholar
Sharma, V., Nand, S., Pramanik, S., Chen, C.-Y. & Mishra, M. 2020 Control of radial miscible viscous fingering. J. Fluid Mech. 884, A16.10.1017/jfm.2019.932CrossRefGoogle Scholar
Speirs, N.B., Mansoor, M.M., Belden, J. & Truscott, T.T. 2019 Water entry of spheres with various contact angles. J. Fluid Mech. 862, R3.10.1017/jfm.2018.985CrossRefGoogle Scholar
Takamura, K., Fischer, H. & Morrow, N.R. 2012 Physical properties of aqueous glycerol solutions. J. Petrol. Sci. Engng 98, 50–60.10.1016/j.petrol.2012.09.003CrossRefGoogle Scholar
Tanaka, Y., Yamazaki, Y. & Okumura, K. 2003 Bouncing gel balls: impact of soft gels onto rigid surface. Europhys. Lett. 63 (1), 146–152.10.1209/epl/i2003-00462-xCrossRefGoogle Scholar
Taylor, G.I. 1950 The formation of a blast wave by a very intense explosion. - II. The atomic explosion of 1945. Proc. R. Soc. Lond. A: Math. Phys. Sci. 201 (1065), 175–186.Google Scholar
Tran, T., de Maleprade, H., Sun, C. & Lohse, D. 2013 Air entrainment during impact of droplets on liquid surfaces. J. Fluid Mech. 726, R3.10.1017/jfm.2013.261CrossRefGoogle Scholar
Truscott, T.T., Epps, B.P. & Belden, J. 2014 Water entry of projectiles. Annu. Rev. Fluid Mech. 46 (1), 355–378.10.1146/annurev-fluid-011212-140753CrossRefGoogle Scholar
Vella, D., Lee, D.-G. & Kim, H.-Y. 2006 The load supported by small floating objects. Langmuir 22 (14), 5979–5981.10.1021/la060606mCrossRefGoogle ScholarPubMed
Vella, D. & Metcalfe, P.D. 2007 Surface tension dominated impact. Phys. Fluids 19 (7), 072108.10.1063/1.2747235CrossRefGoogle Scholar
Watson, D.A., Thornton, M.R., Khan, H.A., Diamco, R.C., Yilmaz-Aydin, D. & Dickerson, A.K. 2024 Water striders are impervious to raindrop collision forces and submerged by collapsing craters. Proc. Natl Acad. Sci. USA 121 (5), e2315667121.10.1073/pnas.2315667121CrossRefGoogle ScholarPubMed
Wu, S. & Shanks, R. 2004 Solubility study of polyacrylamide in polar solvents. J. Appl. Polym. Sci. 93 (3), 1493–1499.10.1002/app.20608CrossRefGoogle Scholar
Ye, X. & Meer, D.V.D. 2021 Hydrogel sphere impact cratering, spreading and bouncing on granular media. J. Fluid Mech. 929, A24.10.1017/jfm.2021.851CrossRefGoogle Scholar
Zhao, H., Brunsvold, A. & Munkejord, S.T. 2011 Transition between coalescence and bouncing of droplets on a deep liquid pool. Intl J. Multiphase Flow 37 (9), 1109–1119.10.1016/j.ijmultiphaseflow.2011.06.007CrossRefGoogle Scholar
Zhao, S.-C., de Jong, R. & van der Meer, D. 2019 Formation of a hidden cavity below droplets impacting on a granular substrate. J. Fluid Mech. 880, 59–72.10.1017/jfm.2019.742CrossRefGoogle Scholar
Supplementary material: File

Ye et al. supplementary material movie 1

A hydrogel sphere with diameter D0 = 4 mm and Young’s modulus Y = 150 Pa impacts a water surface at 1.5 m/s.
Download Ye et al. supplementary material movie 1(File)
File 7.2 MB
Supplementary material: File

Ye et al. supplementary material movie 2

A hydrogel sphere with diameter D0 = 4 mm and Young’s modulus Y = 150 Pa impacts a water surface at 2.5 m/s.
Download Ye et al. supplementary material movie 2(File)
File 7 MB
Supplementary material: File

Ye et al. supplementary material movie 3

A hydrogel sphere with diameter D0 = 4 mm and Young’s modulus Y = 150 Pa impacts a water surface at 3.5 m/s.
Download Ye et al. supplementary material movie 3(File)
File 7.3 MB
Supplementary material: File

Ye et al. supplementary material movie 4

A hydrogel sphere with diameter D0 = 4 mm and Young’s modulus Y = 150 Pa impacts a water surface at 4 m/s.
Download Ye et al. supplementary material movie 4(File)
File 7.4 MB